z z solid, which is in turn an 'equidistance surface'. According to the potential theory of adsorption, the adsorption energy on this surface can be approximated by —kT ln(x). The corresponding surface coverage can be found by calculating the volume within the bounds of the equidistance surface V(z). If one neglects the B'a3 term in Equation (6.30) and normalizes the volume V(z) so that V(a) = Nma3, then

Note that the behavior predicted by FHH theory in the regime N/Nm 1 is physically meaningless because, at small values of z, the repulsive energy term in Equation (6.33) cannot be neglected. The log-log plot of Equation (6.37):

allows direct evaluation of the value of Ds as the slope of a straight line that approximates the experimental data.

The basic assumption underlying the FHH approach is that the potential energy over a fractal surface should depend only on z and it is given by the 'flat surface expression', i.e. by the relation —A/z3. This assumption neglects effects due to the energetic heterogeneity of adsorbing surfaces.

Since the FHH method is relatively simple, it was widely used for determining the surface fractal dimension of several solids, including active carbons [35, 56], aerogels [57], metal films, oxides and related compounds [55]. In particular, Pfeifer and Lui [55] have provided an almost comprehensive review on this topic. Section 6.5 will briefly review applications to soils.

The potential energy in Equation (6.33) assumes that the contributions from several nearby adsorbing surfaces are rare, i.e. that multiple-surface effects are unimportant. This adsorption regime is usually called the 'van der Waals regime'. In the case of adsorption inside a porous network, the surface-molecule attraction is stronger than that predicted by the z—3 power law, due to the 'overlapping surface' effect. In such cases, adsorption is governed by the so-called capillary condensation mechanism. A discussion of an FHH isotherm that obeys this mechanism was presented in [35,55], whereas an outline of the derivation of this isotherm based on [58] is provided below. In the case of adsorption in porous networks, the FHH isotherm was shown to be a generalization of the well-known Dubinin-Radushkevich (DR) equation [59, 60]:

which describes the filling process of a pore whose diameter is R and where MDR is a positive constant. In order to derive the FHH equation, Avnir and Jaroniec [58] considered a network of structurally heterogeneous pores of sizes in the range [Rmn, RmJ.

If the pores are independently filled and the filling of each pore is described by the DR equation, then the overall adsorption isotherm is


Nm J

where J(R) is the pore size distribution. For fractal pores, the size distribution has the form [48, 61]


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